1. Let α be an irrational number. Prove that there exist infinitely many rational
numbers p/q satisfying (2.25).

Let \( \alpha \) be an irrational number. The task is to prove that there exist infinitely many rational numbers \( \frac{p}{q} \) (where \( p \) and \( q \) are integers) satisfying the following inequality:

\[
\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}
\]

This result is a famous theorem from Diophantine approximation known as **Dirichlet's approximation theorem**. The theorem guarantees that for any irrational number \( \alpha \), there are infinitely many rational numbers \( \frac{p}{q} \) such that:

\[
\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}
\]

### Proof (Outline of Dirichlet's Approximation Theorem):

1. **The pigeonhole principle**:  
   Consider the fractional parts \( \{ n\alpha \} \) for \( n = 1, 2, \dots, Q \), where \( \{ x \} \) denotes the fractional part of \( x \). These are values in the interval \( [0, 1) \), and there are \( Q \) such values.

2. **Partition the interval**:  
   Divide the interval \( [0, 1) \) into \( Q \) equal subintervals of length \( \frac{1}{Q} \). By the pigeonhole principle, at least two of the fractional parts \( \{ n_1 \alpha \} \) and \( \{ n_2 \alpha \}

Let α be an irrational number. Prove that there exist infinitely many rational numbers p/q satisfying (2.25).

Explore the proof that for any irrational number α, there are infinitely many rational approximations p/q such that |α - p/q| < 1/q². This proof uses Dirichlet's Approximation Theorem and the pigeonhole principle, suitable for advanced mathematics students.

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